Many computer-implemented displays consist of two-dimensional arrays of individual picture elements, or pixels. To form an image, a display driver selectively colors the pixels. Because the individual pixels are so small, the display appears to a human viewer to be a continuous rendering of an image. This illusion is particularly effective for complex images of continuous tones such as photographs.
For simple geometric shapes, however, the pixelated nature of the display can become apparent to the human viewer. For example, if the display driver is instructed to draw a straight line, there is no guarantee that the points on that desired line will coincide with the pixels that are available for rendering the line. As a result, the desired line is often rendered by a rasterized line of pixels that are close to but not necessarily coincident with the desired line. This results in rasterized lines that have a jagged or echeloned appearance.
In the course of rendering an image, a large number of straight lines and line segments are often drawn. As a result, given a desired line, the display driver must frequently select those pixels that will minimize the jagged appearance of the resulting rasterized line. A straightforward mathematical approach is to use the equation of the desired line and the coordinates of the available pixels to minimize a least square error across all points on the line. While such an approach has the advantage of globally optimizing the selection of pixels on the rasterized line, the large number of floating point operations required causes this approach to be prohibitively time-consuming.
To meet constraints on speed, display drivers typically implement rasterization methods that avoid time-consuming floating point operations. Among the methods that meet the foregoing constraints is that taught in Bresenham, J. E., “Algorithm for Computer Control of a Digital Plotter,” IBM System Journal, Vol. 4, pp.25-30, 1965, the contents of which are herein incorporated by reference. Using only integer operations, the Bresenham algorithm reduces the choice of what pixel to select to an examination of the sign of a discriminant.
Even faster rasterization methods exist that select multiple points on the rasterized line based on the outcome of a single decision. There also exist a variety of rasterization methods aimed at rendering conic sections and quadric sections on a pixelated display.
However, the foregoing rasterization methods all rely on the assumption that the array of pixels is arranged in a uniform rectangular grid that can readily be modeled by a Cartesian coordinate system. This is a reasonable assumption given the prevalence of two-dimensional displays such as computer monitors and printers at the time these algorithms were developed.
Since then, however, volumetric, or three-dimensional displays have been developed. Such displays permit the generation, absorption, or scattering of visible radiation from a set of localized and specified regions within a volume. Examples of such systems are taught in Hirsch U.S. Pat. No. 2,967,905, Ketchpel U.S. Pat. No. 3,140,415, Tsao U.S. Pat. No. 5,754,147 and on pages 66-67 of Aviation Week, Oct. 31, 1960.
In a typical volumetric display 1, shown in FIG. 1, a motor 2 spins an imaging plate 3 rapidly about an axis 4. A light source 5 under the control of a display driver 6 illuminates selected spots 7 on the imaging plate 3 at successive instants. If the imaging plate 3 spins rapidly enough, and if the successive instants are separated by short enough time intervals, a continuous curve will appear to hang in mid-air.
FIG. 2 illustrates an example in which the display driver 6 renders a line 8 in a plane perpendicular to the axis 4. In FIG. 2, the imaging plate 3 is shown in six successive instants as it rotates around the axis 4, now perpendicular to the page. At each of the six instants, the light source 5, under the control of the display driver 6, illuminates a spot 7 on the imaging plate 3. As shown in FIG. 2, by aiming the light source 5 at the correct spot and firing the light source 5 at the right time, it is possible to trace out the line 8. It is the function of the display driver 6 to correctly aim and fire the light source 5 so as to trace out the line 8.
To aim the light source 5, the display driver 6 needs a way to identify points in space. In other words, the display driver 6 needs a coordinate system. One possible coordinate system is a Cartesian coordinate system.
Using a Cartesian coordinate system, the display driver 6 would draw the line 8 in FIG. 2 by specifying, for example, that at time t1 the light-source 5 should aim 30 units north, at time t2, the light-source 5 should aim 29 units north and 1 unit east, at time t3, the light-source 5 should aim 28 units north and 2 units east, and so on. To ensure that the imaging plate 3 is at the appropriate location, the display driver 6 also has to specify the values of the times t1, t2 and t3 based on how fast the imaging plate 3 is spinning. Although the spinning of the imaging plate 3 can be resolved into a north-south component and an east-west component, this is a computationally taxing exercise that can easily be avoided by using a polar coordinate system.
As an alternative, the display driver 6 could draw the line in FIG. 2 in polar coordinates by specifying, for example, that at time t1, the light source 5 should aim 30 units away from the axis 4 at a 90 degree angle, at time t2, the light source 5 should aim 29.02 units from the axis 4 at an angle of 88.03 degrees, and at time t3, the light source 5 should aim at 28.07 units from the axis 4 at an angle of 85.91 degrees. This would, of course, draw the same line that was drawn using Cartesian coordinates. The advantage of using polar coordinates becomes apparent when it comes time to specify when to fire the light source 5. Because the imaging plate 3 is spinning, it is far more natural to represent its motion in terms of degrees per second than it is to resolve its motion into a north-south component and an east-west component. For this reason, calculations involving motion of the imaging plate 3 in a volumetric display are best performed in terms of a polar coordinate system, or in a three-dimensional case, a cylindrical coordinate system.
The process of rendering a line in a polar coordinate system is fundamentally different from that of rendering a line in a Cartesian coordinate system. In a Cartesian coordinate system, a line has a constant slope. A unit change in the horizontal direction always results in the same change in the vertical direction, regardless of where that change occurs. This is not the case in a polar coordinate system.
In a polar coordinate system, the slope of a line can vary dramatically with position along the line. In contrast to a Cartesian coordinate system in which the slope of a line is constant at all points on the line, the slope of a line in a polar coordinate system can vary all the way from minus infinity to positive infinity, passing through zero in the process. For those portions of the line that are closest to the origin of the polar coordinate system, the change in distance of the line from the origin changes very slowly with angle. In these portions, the slope of the line is close to zero. For those portions of the line that are far from the origin of the polar coordinate system, even a small change in angle results in a huge change in radial distance from the origin. For these portions of the line, the slope approaches infinity.